Math: 466b87e3

ID: 466b87e3 

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)

Comment: If you understand the principles behind systems of equations having no solution or infinite solutions, this question should take about 5 seconds.

Method 1: Because there is no manipulation required of the two equations provided—the coefficient for y, 1, is the same in both, it must be true that c = 1/2 for the system to have no solution, since it is impossible for y = (1/2)x plus or minus whatever to intersect another graph that starts with y = (1/2)x, provided the part that follows is not identical, of course, in which case the line is the same, and there would be infinite solutions. In short, same y, same slope, different detached number (c term), no solution. (Same y, same slope, same detached number (c term), infinite solutions.)

Method 2: You could use Desmos and add a slider for c to make sure you had the correct answer. Look for where the lines become parallel.

Is it wrong to take a minute on a supposedly Hard (level 3) question? No. All that matters is that you get the correct answer within a reasonable amount of time. Clearly, the correct answer is 0.5 or 1/2.

Method 3: If either of the two methods above eluded you, but you felt pretty comfortable with the algebra, you could solve using substitution or elimination. Here, I might substitute, since both equations are set to y. 

y = y

(cx + 10) = ((1/2)x + 8)

Now, if c = 1/2, there will clearly be a problem.

((1/2)x + 10) = ((1/2)x + 8)

10 = 8 X

Alternatively, you could eliminate:

y = (1/2)x + 8

y = cx + 10

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0 = (1/2)x - cx - 2

2 = (1/2)x - cx

Because there are two unknowns, c and x, the only way to guarantee that no value of x could make the right-hand side equal 2 is to set c equal to 1/2, since that would wipe out the right-hand side altogether, and 2 ≠ 0. Thus, c must be 1/2 or 0.5.

These solutions still build off the fundamental understanding that whether a system of equations has no solution or an infinite number of solutions, the coefficients for each of x and y will match perfectly. The term that has no unknown attached to it, the c term, will determine whether the system has no solution or an infinite number of solutions.